Logistic regression is one of the most popular models to classify in data science, and in general, it is easy to use. However, in order to conduct a goodness-of-fit test, we cannot apply asymptotic methods if we have sparse datasets. In the case, we have to conduct an exact conditional inference via a sampler, such as Markov Chain Monte Carlo (MCMC) or Sequential Importance Sampling (SIS). In this chapter, the authors investigate the rejection rate of the SIS procedure on a multiple logistic regression models with categorical covariates. Using tools from algebra, they show that in general SIS can have a very high rejection rate even though we apply Linear Integer Programming (IP) to compute the support of the marginal distribution for each variable. More specifically, the semigroup generated by the columns of the design matrix for a multiple logistic regression has infinitely many “holes.” They end with application of a hybrid scheme of MCMC and SIS to NUN study data on Alzheimer disease study.
